Abstract
An \(\alpha ,\beta \)-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors \(\alpha \) and \(\beta \). Two k-colorings of a graph are k-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than k colors). Las Vergnas and Meyniel showed that if a graph is \((k-1)\)-degenerate, then each pair of its k-colorings are k-Kempe equivalent. Mohar conjectured the same conclusion for connected k-regular graphs. This was proved for \(k=3\) by Feghali, Johnson, and Paulusma (with a single exception \(K_2\square \,K_3\), also called the 3-prism) and for \(k\ge 4\) by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment L and an L-coloring \(\varphi \), a Kempe swap is called L-valid for \(\varphi \) if performing the Kempe swap yields another L-coloring. Two L-colorings are called L-equivalent if we can form one from the other by a sequence of L-valid Kempe swaps. Let G be a connected k-regular graph with \(k\ge 3\) and \(G\ne K_{k+1}\). We prove that if L is a k-assignment, then all L-colorings are L-equivalent (again excluding only \(K_2\square \,K_3\)). Further, if \(G\in \{K_{k+1},K_2\square \,K_3\}\), L is a \(\Delta \)-assignment, but L is not identical everywhere, then all L-colorings of G are L-equivalent. When \(k\ge 4\), the proof is completely self-contained, implying an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let H be a graph such that for every degree-assignment \(L_H\) all \(L_H\)-colorings are \(L_H\)-equivalent. If G is a connected graph that contains H as an induced subgraph, then for every degree-assignment \(L_G\) for G all \(L_G\)-colorings are \(L_G\)-equivalent.
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This is because in a line graph each subgraph recolored by a Kempe swap must be a path or an even length cycle. So it is easier to understand how the coloring changes after a single Kempe swap.
By requiring at least one L-coloring, we can state many of our results more cleanly.
The general case of this result easily reduces to the case when G is 2-connected, which is known as Rubin’s Block Lemma. For a shorter proof, see Section 9 of [4].
To prove this analogue, we greedily color the vertices of \(G\setminus H\) in order of non-increasing distance from H. Afterward, we can extend this coloring to H precisely because H is degree-choosable.
Recall that Rubin’s Block Lemma [6] says a block contains such a cycle if and only if it is non-Gallai.
Although we have not yet proved this lemma, its proof is independent of the present theorem, so invoking it now is logically consistent. We make this choice to preserve the narrative flow of Sect. 4.
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Thanks to a referee for carefully reading this manuscript and making numerous suggestions that helped improve the presentation.
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Dedicated to the memory of Landon Rabern.
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Cranston, D.W., Mahmoud, R. Kempe Equivalent List Colorings. Combinatorica 44, 125–153 (2024). https://doi.org/10.1007/s00493-023-00063-2
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DOI: https://doi.org/10.1007/s00493-023-00063-2